The book said that you could substitute 묻다 with 끼다 to give a more grime-encrusted impression. Here is how the book explained it in Korean:
'때가 끼다'라고 하면 '묻다' 보다 더 오랫동안 오물이 계속 쌓여서 잘 떨어지지 않을 정도로 엉겨붙는 것을 말하다.Notice that the writer wrote 엉겨붙는 것을, which means "solidifying" rather than "solidified" (엉겨붙은 것을). In other words, the writer wrote that the grime was still in the process of solidifying rather than already in a state of solidification. Is that what the writer meant to say? I do not think so.
If you say 때가 끼다, you are saying that the grime has collected over a longer period of time than 때가 묵다 and is so solidified that it does not come off easily.
I think the writer meant to say "solidified," which means he should have written it either as 엉겨붙은 것을 or 엉겨붙어 있는 것을.